Difference between revisions of "2007 AMC 8 Problems/Problem 16"

Revision as of 20:33, 6 January 2018

Problem

Amanda draws five circles with radii $1, 2, 3, 4$ and $5$ . Then for each circle she plots the point $(C,A)$ , where $C$ is its circumference and $A$ is its area. Which of the following could be her graph?

$\textbf{(A)}$ $[asy] size(75); pair A= (1.5,2) , B= (3,4) , C= (4.5,7) , D= (6,11) , E= (7.5,16) ; draw((0,-1)--(0,16)); draw((-1,0)--(16,0)); dot(A^^B^^C^^D^^E); label("$A$", (0,8), W); label("$C$", (8,0), S);[/asy]$

$\textbf{(B)}$ $[asy] size(75); pair A= (1.5,9) , B= (3,6) , C= (4.5,6) , D= (6,9) , E= (7.5,15) ; draw((0,-1)--(0,16)); draw((-1,0)--(16,0)); dot(A^^B^^C^^D^^E); label("$A$", (0,8), W); label("$C$", (8,0), S);[/asy]$

$\textbf{(C)}$ $[asy] size(75); pair A= (1.5,2) , B= (3,6) , C= (4.5,8) , D= (6,6) , E= (7.5,2) ; draw((0,-1)--(0,16)); draw((-1,0)--(16,0)); dot(A^^B^^C^^D^^E); label("$A$", (0,8), W); label("$C$", (8,0), S);[/asy]$

$\textbf{(D)}$ $[asy] size(75); pair A= (1.5,2) , B= (3,5) , C= (4.5,8) , D= (6,11) , E= (7.5,14) ; draw((0,-1)--(0,16)); draw((-1,0)--(16,0)); dot(A^^B^^C^^D^^E); label("$A$", (0,8), W); label("$C$", (8,0), S);[/asy]$

$\textbf{(E)}$ $[asy] size(75); pair A= (1.5,15) , B= (3,10) , C= (4.5,6) , D= (6,3) , E= (7.5,1) ; draw((0,-1)--(0,16)); draw((-1,0)--(16,0)); dot(A^^B^^C^^D^^E); label("$A$", (0,8), W); label("$C$", (8,0), S);[/asy]$

Solution

The circumference of a circle is obtained by simply multiplying the radius by $2\pi$ . So, the C-coordinate (in this case, it is the x-coordinate) will increase at a steady rate. The area, however, is obtained by squaring the radius and multiplying it by $\pi$ . Since squares do not increase in an evenly spaced arithmetic sequence, the increase in the A-coordinates ( aka the y- coordinates) will be much more significant. The answer is $\boxed{\textbf{(A)}},$ \textbf{(A)} $ $[asy] size(75); pair A= (1.5,2) , B= (3,4) , C= (4.5,7) , D= (6,11) , E= (7.5,16) ; draw((0,-1)--(0,16)); draw((-1,0)--(16,0)); dot(A^^B^^C^^D^^E); label("$A$", (0,8), W); label("$C$", (8,0), S);[/asy]$ $.

2007 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 15	Followed by Problem 17
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Difference between revisions of "2007 AMC 8 Problems/Problem 16"

Revision as of 20:33, 6 January 2018

Problem

Solution

See Also

@@ Line 77: / Line 77: @@
 == Solution ==
-The circumference of a circle is obtained by simply multiplying the radius by <math>2\pi</math>. So, the C-coordinate (in this case, it is the x-coordinate) will increase at a steady rate. The area, however, is obtained by squaring the radius and multiplying it by <math>\pi</math>. Since squares do not increase in an evenly spaced arithmetic sequence, the increase in the A-coordinates ( aka the y- coordinates) will be much more significant. The answer is <math>\boxed{\textbf{(A)}}</math>
+The circumference of a circle is obtained by simply multiplying the radius by <math>2\pi</math>. So, the C-coordinate (in this case, it is the x-coordinate) will increase at a steady rate. The area, however, is obtained by squaring the radius and multiplying it by <math>\pi</math>. Since squares do not increase in an evenly spaced arithmetic sequence, the increase in the A-coordinates ( aka the y- coordinates) will be much more significant. The answer is <math>\boxed{\textbf{(A)}},
+</math> \textbf{(A)} $
+<asy>
+size(75);
+pair A= (1.5,2) ,
+B= (3,4) ,
+C= (4.5,7) ,
+D= (6,11) ,
+E= (7.5,16) ;
+draw((0,-1)--(0,16));
+draw((-1,0)--(16,0));
+dot(A^^B^^C^^D^^E);
+label("$A$", (0,8), W);
+label("$C$", (8,0), S);</asy>$.
 ==See Also==
 {{AMC8 box|year=2007|num-b=15|num-a=17}}
 {{MAA Notice}}